Analytic theory of stochastic oscillations in single-cell gene expression

Noise-induced oscillations in individual cells are usually characterized by a non-monotonic power spectrum with an oscillatory autocorrelation function. Here we develop an analytical approach of stochastic oscillations in a minimal stochastic gene expression model including promoter state switching, protein synthesis and degradation, as well as a genetic feedback loop. The autocorrelated function, power spectrum, and probability distribution of protein concentration fluctuations are computed in closed form. Using the solvable model, we illustrate oscillations as a stochastic circulation along a hysteresis loop. A triphasic stochastic bifurcation upon the increasing strength of negative feedback is observed, which reveals how stochastic bursts evolve into stochastic oscillations. Translational bursting is found to enhance the robustness and broaden the region of stochastic oscillations in our model. These results provide deeper insights into R. Thomas' two conjectures for single-cell stochastic gene expression kinetics.